We apply the method of Lyapunov functions. As a Lyapunov function, take the following infinitely differentiable function \(V : \mathbb {R} \rightarrow [0, \infty )\) :
\(V(c) := {\left\lbrace \begin{array}{ll} |c|,\, c \ge 2;\\ 0,\, c \le 1.\end{array}\right.}\) 
This function can be constructed by smoothing kernel and convolution. The generator for each \(C_k\)  in (REF ) is given by
\(\mathcal {L}f(c) := \tilde{\gamma }(c)f^{\prime }(c) + \frac{1}{2}\tilde{\sigma }^2(c)f^{\prime \prime }(c).\) 
Recall (REF ): \(\varlimsup \limits _{c \rightarrow \infty }\tilde{\gamma }(c) < 0\)  and \(\varliminf \limits _{c \rightarrow -\infty }\tilde{\gamma }(c) > 0\) . Combining this with (REF ), we get:
\(\varlimsup \limits _{|c| \rightarrow \infty }\mathcal {L}V(c) < 0.\) 
From the articles [1]} and [2]}, we get tightness of each \(C_k\) . That is, \(\sup _{t \ge 0}\mathbb {P}(|C_k(t)| \ge c) \rightarrow 0\)  as \(c \rightarrow \infty \) . The same is true for the vector \(\mathbf {C} = (C_1, \ldots , C_n)\) . Combining this observation with the following property: for each \(i\) ,
\(\mathbb {P}(a < C_i(t) < b\mid C_i(0) = x) > 0,\quad x, a, b \in \mathbb {R}\) , we complete the proof.
